Cheep-Cheeps. Unlike virtually every other enemy seen in the franchise, they respawn constantly and their movement makes them difficult to avoid by simply running fast. And like every other enemy, they drain your life or kill you with any touch that isn't a Goomba Stomp.
Bloopers/Bloobers in 1 and Lost Levels. Especially when they decide to pop up in the middle of an overground level for no reason. Squids in mid-air!
Super Mario Bros. 2 had Phanto. He really doesn't like you taking keys, and flies around harassing you until you can finally get to that locked door.
There's also the Angry Sun Super Mario Bros. 3, which flies erratically all around the screen. Fortunately, it only appears twice in the entire game, and can be killed for good for the rest of the level provided you have the ultra-useful Hammer Suit or hit it with a Koopa shell.
Everything in Super Mario Land is a pain in the arse by world three. Especially those flying head things that always bounce around on a series of platforms that you have to jump onto. Also, in any Mario game, Spiders manage to be a pain whenever they appear.
Super Mario Galaxy has literal Goddamned Bats in a few levels. Usually, they just bump into you and are a pain in the ass to kill. But put them in the Slippy-Slidey Ice World and you get Bats that freeze you solid, and can only be killed by the temporary power-up that is difficult to aim at stationary objects, let alone flying ones. What really makes them irritating is that if they score a hit, they actually laugh at you.
Lakitu. The bastards fly around, always just out of reach, while dropping spikes into the player. Peach help anyone on the receiving end of a Lakitu that drops independent enemies.
Goobles, which constantly spawn out of the goop Mario has been tasked with cleaning the plaza and island in general of when he comes near it. Should they miss Mario with is attacks or simply be allowed to wander around for too long, they will produce more goop, which can in turn spawn more of these goopy Goobles.
Wind Spirits, which most commonly appear to award you for reaching a high place by knocking you back down, homing in on Mario and only leaving him alone if you voluntarily descend back to a lower level. They can be eaten with Yoshi, but anything else you hit them with will only stop them temporarily at best, and since Pianta Village is in the mountains you don't even have the option of voluntarily dropping down to get rid of them.
Bees are only ever encounter near things the player wants, such as fruit or coins, or on narrow ledges that are easy to fall off of when the player is doing something like trying to avoid being stung by bees. Pianta Village combines these by putting a beehive next to a red coin on the edge of a cliff! Eating bees with Yoshi is the only way to combat them, and if you want blue coins, you must not only eat every bee outside of the hives but then knock the hives down and eat every bee that understandably comes after you. They're tiny, and thus harder to see as their numbers decrease.
Poinks in any instance other than the second Petey Piranha confrontation where you want to use the squirt nozzle, as they are attracted to Mario like moths to a flame, almost always come in large swarms and will force you to waste water getting rid of them if one manages to touch Mario. And they tend to respawn.
Coo Coos, which fly in erratic patterns well above Mario's jump height dropping Gooble-spawning goop. The FLUDD squirt nozzle can lower them to ground level, but this is a slow process that relies on constantly spraying one, and there is almost always more than one.
Cataquacks are fairly non-alert, sometimes falling asleep. To make up for this, they are all over Gelato Beach and will chase Mario very far if they do notice him. Blue ones are annoying enough, as it is very easily to get continuously tossed into the air by them with no option other than waiting it out if one manages to catch Mario once, but red ones are worse, as they damage Mario with each toss. To get rid of them, they have to be lured into the ocean, lured near a sprouting dune bud, or eaten with Yoshi, all of which one has to go out of their way to do. And the final mission of Gelato Beach requires you to push watermelons through the level, which Cataquacks can and will easily smash to pieces, requiring you to use up most of your water trying to hold them at bay.
A game called Pharaoh's Curse for the Commodore 64 has a very pesky bat (or is it a bird?). It doesn't kill you at all. No, what it does is swoop down, pick you up, and carry you to a random spot, forcing you to drop any key you were carrying, leaving yourself unable to open the locked door you were planning to. Of course, you can only shoot sideways, so you would be often helpless to avoid it when it swooped down from above at high speed.
Some segments of Viewtiful Joe feature annoying swarms of infinite robotic bats that stick to you, stunning you, and explode unless you button mash them away. They make you solve puzzles slower, and allow other enemies to hit you, but can be fended off with a Hurricane Kick.
Robo Birds in Wario Land 4, found only in Hard mode and above, pretty much cover this. They fly in the air right near platforming sections, and charge at Wario when he gets near. Not hard to defeat (one hit kills it), but considering that it's found in narrow spaces, doesn't give you anything for killing it, and respawns when you re enter the area, even though every other non transformation enemy doesn't come back unless you win/lose the level, it's freaking annoying. Plus, Pinball Zone (yes, that Pinball Zone that named the trope) is annoying enough without them.
Trine has literal bats that: Always appear at the worst moment (like when you're trying to avoid falling into a Bottomless Pit), always travel in packs, are hard to hit (only one of the characters has an air attack) and deplete your Life Meter very rapidly.
Green and yellow Klobbers in Donkey Kong Country 2: Diddy's Kong Quest, which only bump you around (and take bananas in the case of the yellow ones). They're rarely much of a threat, but they're often very annoying especially in Klobber Karnage.
The Swoopys in Donkey Kong Country 3: Dixie Kong's Double Trouble!. Little woodpeckers that you think are a Unique Enemy... until you reach the Lost World, where they get a level all to themselves. They fly across the stage in never-ending queues; killing one can make a gap in the line, but there will always be more. Most of the level you have to fly past them as Squawks the parrot, with squirrelly controls and a hard-to-aim arcing projectile attack. To reach the bonus levels, you have to slip through the knotholes they use, going with them once, going against them another. And they're in the bonus levels too, resulting in failure if one hits you!
Zingers dive bomb the player and drop grape grenades. They can be shot out of the sky, but they just respawn shortly. Goddamned Bees...
Purple Klaptraps. These things will chase you down if they see you, and are immune to all standard attacks. There are two ways to kill them: orange bombs and your musical instrument attack. The former is horribly inaccurate, and can actually hurt you if you aim it wrong. The latter is the rough equivalent of using a nuke to kill a squirrel. The fact that these things don't drop health when you kill them does not help.
The starfish enemy Shuri, mostly because you're defenseless underwater where they live. This becomes more obvious in the Splish Splash Salvage minigame, where they cause you to instantly lose if hit by them.
The crabs in the beach levels, due to taking too many steps to defeat for how common they are: first, you must jump on them, making them lift their pincers and makes them Spinies. Then, you must hit the ground to flip them over (if you do this before jumping on them, they use their pincers to stop themselves from flipping over), then you can jump on them and finish them off.
Crystal Caves has the birds (in underground caverns??) who fly high up, beyond the reach of your horizontally-shooting gun, and bombard you with hard-to-dodge eggs. And the eggs remain on the ground for a while after they fall and crack, and can harm you when you step on them. And sometimes they hatch into another, smaller bird when on the way down.
Super Mario World 2: Yoshi's Island has Grinders, monkeys that plague World 3 and are a real pest. They're fast, take multiple hits and have several methods of annoying you; such as grabbing Baby Mario right off Yoshi's back and running off, spitting seeds at you and dropping bombs, which make holes in the level, making it easy for you to fall to your death. You will want to go to the nearest jungle and exterminate every monkey you can find by the time you're done with World 3.
The Snuggle(?) Bunnies in Nicktoons: Attack Of The Toybots. They spawn in about groups of two or more, are small and difficult to hit, and once they see you they'll try and cling to you, rendering you unable to attack. If you don't managed to shake them off in time they'll explode, flinging you merrily into the air, and often off a platform.
Bug! With six worlds, each world is bound to have one:
Insectia: Bees. NOT THE BEES! The first Airborne Mook that Bug will encounter, they will try to swoop in on Bug at an awkward angle, and Bug will have to time his jump exactly in order to beat them. Some of them even come in from nowhere, just when you thought it was safe!
Reptilia: Army ants. As in, ants that drop via parachutes, wear army hats, and fire bouncy grenades from their tails. The main problem is that they tend to be positioned at the top of an incline, makign it hard to dodge the grenades.
Splot!: Frogs, frogs, and more frogs. These things jump quite quickly, take three hits to die, and their tongue attack has good range. There are also literal lightning bugs, another Airborne Mook that attacks from an awkward angle to hit and fires out a stream of lightning at the floor.
Quaria: Sea worms move quite fast (especially when they start rolling when you get near) and take five hits to die. Fire-breathing moray eels are also very annoying because Bug starts out so far from them and will have to get closer while avoiding their fireballs.
Burrbs: Snow fleas. Some just jump out of the floor here and there. Some jump out of the floor and throw snow at Bug. Some stay at one point rolling snowballs at Bug until Bug gets near, at which they run away. And the worst kind are those who assault Bug with snowballs from a distance!
Arachnia: Green Flies that vomit a pool of acid on the ground. They come in large numbers from offscreen. Also, there are spiders who roll bowling balls at Bug. Not too bad on level ground, but they usually do it from the top of a slope...
Spyro: A Hero's Tail has some literal Goddamned Bats in the Dragonfly Falls level - rest assured that if you enter a cave or tunnel at any time in the level, several bats will fly out. They deal damage just by touching you, and unless you're psychic and know they're coming, you'll be too late to kill them every time as they just fly away after taking half your health. It's a prime example of a trap for beginners and first time players of the game. Plus to add insult and to piss the player off even more they never respawn like the rest of the enemies in the level so you can't get revenge on them. Oh and charging them will make you take damage even though it kills them. Bullshit!
The goddamned Bushrats. They're almost impossible to jump on due to their size, they throw spears at you, they can cause you to accidentally spin away health powerups in the process of killing them, and they can also cause you to spin off a platform and into oblivion.
The actual bats in the ruins levels. They can camouflage themselves against the dark background to make themselves harder to see.
Mario vs. Donkey Kong has certain levels that feature an invulnerable bird that flies back and forth and drops eggs. If the bird or one of its eggs touches Mario, he loses a life, forcing you to restart the level from the beginning, and unlike other Mario games, the bird can not be defeated by jumping on it. The levels it appears in are considerably more frustrating than they reasonably should be.
In Atlantis no Nazo, Batabats fly so high the player can't even touch them and throw down triangle-shaped droppings. They're the first enemy encountered in the game.
Any ranged enemy can qualify. Jumping for that jeweled skull? I don't think so, enjoy your damage stun which will send you plummeting into spikes.
Seagulls. Attack you from various places (usually in places where your jump accuracy needs to be perfect or near-perfect), can see you with the invisibility powerup and in 9 cases out of 10 give you nothing in return.
Crabs of bomb-throwing variety. Not only they pop out of spawns at the worst possible moment, their bombs have quite a range. While not damaging, they can still distract you enough to lose bonus points/more hitpoints/even a life. Finally, when hiding in their shells they're invulnerable to anything short of a Magic Claw.
Fishes are like seagulls, but underwater.
And finally, Chameleons, which could count as Demonic Spiders at first. They make themselves invisible, which makes them immune to all damage, and they have a dickish tendency to re-appear right on top of you, causing Collision Damage, followed by an attack which will cause you to lose another 10 hit points.
Crescent Pale Mist crosses this with Demonic Spiders, as the game enemies come in great numbers, hit hard, and tend to lock Yunou in a series of nearly inescapable attacks. One particular area of Chapter 4 is littered with many, many Demonik Blades and a Shuvalzards, among the bunch. Another area in Chapter 4 houses a boss enemy known as Gilgamesh (a golden Magic Armor with powerful attacks), Dark Armors, and Gargoyles. The worst part: You are trapped until the enemies are killed.
Adventure: In certain difficulty levels, a bat flies throughout the game, stealing whatever item the player is carrying and exchanging it for another, or sometimes just outright taking it. This can prove especially annoying as the bat will sometimes steal the player's sword, and replace it with, oh, say, a DRAGON.
Cave Run 3D has these annoying groups of purple bats which, while they don't actually harm you, crowd around you and slow you down just enough for your character to have a higher chance of missing the next jump and going splat.
In level three of 'Sparkster'' for the SNES, on your way to the robot caterpillar boss, you have to make it past robot bats (presumably the same ones from the original game on the Sega Genesis/Mega Drive) without getting hit when being launched from one robot ostrich to the one you battle the robot caterpillar with. As if that weren't enough, just before you fight the robot caterpillar, you have to avoid bomb-throwing robot monkeys (destroying them is optional). Both the robot bats and the robot monkeys are guaranteed to take away at least half a heart. When you do finally make it to the robot caterpillar, you're lucky if you have enough health left to survive while fighting it.
The roguelite platformer Downwell has literal bats that cling to walls until you're below them, then flutter slowly towards you. The problem is that this is a game where you only go down, and your weapons also only fire downwards. By themselves, they're fairly easy to dispatch by waiting and jumping on them, but mix in other enemies and they become a nightmare.